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American Mathematical Monthly Contents—February 2012

In February’s American Mathematical Monthly, we will use convex analysis to solve the generalized Heron problem, study Jacobi sums using linear algebra, learn some interesting properties of Alcuin’s sequence, and consider a continuous analogue of the classic problem of stacking identical bricks to construct a tower of maximal overhang. Our notes consider Riemann maps, the use of power series to prove inequalities, Chebyshev maps of finite fields, and a trigonometric proof of the Collapsing Walls Theorem. Our book review gives an in depth view of John Stillwell’s Road to Infinity: The Mathematics of Truth and Proof, and as always, the Monthly brings you the best in stimulating and challenging problems. —Scott Chapman

Vol. 119, No. 2, pp.87-176.

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ARTICLES

Solving a Generalized Heron Problem by Means of Convex Analysis

Boris S. Mordukhovich, Nguyen Mau Nam, and Juan Salinas Jr.

The classical Heron problem states: on a given straight line in the plane, find a point $$C$$ such that the sum of the distances from $$C$$ to the given points $$A$$ and $$B$$ is minimal. This problem can be solved using standard geometry or differential calculus. In the light of modern convex analysis, we are able to investigate more general versions of this problem. In this paper we propose and solve the following problem: on a given nonempty closed convex subset of $$\mathbb{R}^{s}$$ , find a point such that the sum of the distances from that point to $$n$$ given nonempty closed convex subsets of $$\mathbb{R}^{s}$$ is minimal.

Jacobi Sum Matrices

Sam Vandervelde

In this article we identify several beautiful properties of Jacobi sums that become evident when these numbers are organized as a matrix and studied via the tools of linear algebra. In the process we reconsider a convention employed in computing Jacobi sum values by illustrating how these properties become less elegant or disappear entirely when the standard definition for Jacobi sums is utilized. We conclude with a conjecture regarding polynomials that factor in an unexpected manner.

Alcuin's Sequence

Donald J. Bindner and Martin Erickson

Alcuin of York (c. 740–804) lived over four hundred years before Fibonacci. Like Fibonacci, Alcuin has a sequence of integers named after him. Although not as well-known as the Fibonacci sequence, Alcuin’s sequence has several interesting properties. The purposes of this note are to acquaint the reader with Alcuin’s sequence, to give the simplest available proofs of various formulas for Alcuin’s sequence, and to showcase a new discovery about the period of Alcuin’s sequence modulo a fixed integer.

Chebyshev Mappings of Finite Fields

Julian Rosen, Zachary Scherr, Benjamin Weiss, and Michael E. Zieve

For a fixed prime $$p$$, we consider the set of maps $$\mathbb{Z}/p\mathbb{Z}\rightarrow\mathbb{Z}/p\mathbb{Z}$$ of the form $$a\mapsto T_{n}(a)$$ where $$T_{n}(x)$$ is the degree-$$n$$ Chebyshev polynomial of the first kind. We observe that these maps form a semigroup, and we determine its size and structure.

Collapsing Walls Theorem

Igor Pak and Rom Pinchasi

Let $$P\subset\mathbb{R}^{3}$$ be a pyramid with the base a convex polygon $$Q$$. We show that when other faces are collapsed (rotated around the edges onto the plane spanned by $$Q$$), they cover the whole base $$Q$$.

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